Computing device and method for enforcing passivity of scattering parameter equivalent circuit

ABSTRACT

A computing device and a method for scattering parameter equivalent circuit reads a scattering parameter file from a storage device. A non-common-pole rational function of the scattering parameters in the scattering parameter file is created by applying a vector fitting algorithm to the scattering parameters. Passivity of the non-common-pole rational function is enforced if the non-common-pole rational function does not satisfy a determined passivity requirement.

BACKGROUND

1. Technical Field

Embodiments of the present disclosure relates to circuit simulatingmethods, and more particularly, to a computing device and a method forenforcing passivity of scattering parameter (S-parameter) equivalentcircuit.

2. Description of Related Art

Scattering parameters (S-parameters) are useful for analyzing behaviourof circuits without regard to detailed components of the circuits. TheS-parameters may be measured at ports of a circuit at different signalfrequencies. In a high frequency and microwave circuit design, theS-parameters of the circuit may be used to create a rational function,and the rational function may be used to generate an equivalent circuitmodel, which may be applied to time-domain analysis for the circuitdesign. For judging whether the circuit design satisfies stabilityrequirements, the time-domain analysis result should be convergent. Toensure constringency, the rational function and the equivalent circuitmodel of the S-parameters are required to be passive.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of one embodiment of a computing device forenforcing passivity of S-parameter equivalent circuit.

FIG. 2 is a block diagram of one embodiment of function modules of apassivity enforcing unit in the computing device of FIG. 1.

FIG. 3 is a flowchart of one embodiment of a method for enforcingpassivity of an S-parameter equivalent circuit.

FIG. 4 is a flowchart of one embodiment detailing S12 in FIG. 3.

DETAILED DESCRIPTION

In general, the word “module,” as used hereinafter, refers to logicembodied in hardware or firmware, or to a collection of softwareinstructions, written in a programming language, such as, for example,Java, C, or Assembly. One or more software instructions in the modulesmay be embedded in firmware. It will be appreciated that modules maycomprised connected logic units, such as gates and flip-flops, and maycomprise programmable units, such as programmable gate arrays orprocessors. The modules described herein may be implemented as eithersoftware and/or hardware modules and may be stored in any type ofcomputer-readable medium or other computer storage device.

FIG. 1 is a block diagram of one embodiment of a computing device 30 forenforcing passivity of an S-parameter equivalent circuit. The computingdevice 30 is connected to a measurement device 20. The measurementdevice 20 is operable to measure S-parameters at ports of a circuit 10,to obtain an S-parameter file 32, and stores the S-parameter file 32 ina storage device 34 of the computing device 30. Depending on theembodiment, the storage device 34 may be a smart media card, a securedigital card, or a compact flash card. The measurement device 20 may bea network analyzer. The computing device 30 may be a personal computer,or a server, for example.

In this embodiment, the computing device 30 further includes a passivityenforcing unit 31 and at least one processor 33. The passivity enforcingunit 31 includes a number of function modules (depicted in FIG. 2) Thefunction modules may comprise computerized code in the form of one ormore programs that are stored in the storage device 34. The computerizedcode includes instructions that are executed by the at least oneprocessor 33, to create a non-common-pole rational function of theS-parameters, and analyze if the non-common-pole rational functionsatisfies determined passivity requirements, and enforces passivity ofthe non-common-pole rational function if the non-common-pole rationalfunction does not satisfy the determined passivity requirements.

FIG. 2 is a block diagram of one embodiment of the function modules ofthe passivity enforcing unit 31 in the computing device 30 of FIG. 1. Inone embodiment, the passivity enforcing unit 31 includes a parameterreading module 311, a vector fitting module 312, a passivity analysismodule 313, a passivity enforcing module 314, and an equivalent circuitgeneration module 315. A description of functions of the modules 311 to315 are included in the following description of FIG. 3.

FIG. 3 is a flowchart of one embodiment of a method for enforcingpassivity of an S-parameter equivalent circuit. Depending on theembodiment, additional blocks may be added, others removed, and theordering of the blocks may be changed.

In block S10, the measurement device 20 measures S-parameters at portsof the circuit 10, to obtain the S-parameter file 32, and stores theS-parameter file 32 in the storage device 34. The parameter readingmodule 311 reads the S-parameters from the S-parameter file 32.

In block S11, the vector fitting module 312 creates a non-common-polerational function of the S-parameters by applying a vector fittingalgorithm to the S-parameters. In one embodiment, the non-common-polerational function of the S-parameters may be the following functionlabeled (1):

$\begin{matrix}{{{{S(s)} \approx {\hat{S}(s)}} = \begin{bmatrix}{{\hat{S}}_{11}(s)} & {{\hat{S}}_{12}(s)} & \ldots & {{\hat{S}}_{1N}(s)} \\{{\hat{S}}_{21}(s)} & {{\hat{S}}_{22}(s)} & \ldots & {{\hat{S}}_{2N}(s)} \\\ldots & \ldots & \ldots & \ldots \\{{\hat{S}}_{N\; 1}(s)} & {{\hat{S}}_{N\; 2}(s)} & \ldots & {{\hat{S}}_{NN}(s)}\end{bmatrix}},{wherein},{{{\hat{S}}_{ij}(s)} = {\sum\limits_{m = 1}^{M}{\frac{r_{m}^{i,j}}{s + p_{m}^{i,j}}.}}}} & (1)\end{matrix}$

In the function (1), M represents control precision of the function, Nrepresents the number of the ports of the circuit 10, r_(m) representsresidue values, p_(m) represents pole values, s=ω=2πƒ, ƒ represents afrequency of a test signal, and d_(m) represents a constant. It isunderstood that control precision means how many pairs of pole-residuevalues are utilized in the function (1).

In block S12, the passivity analysis module 313 determines if thenon-common-pole rational function satisfies a determined passivityrequirement. A detailed description of block S12 is depicted in FIG. 4.If the non-common-pole rational function does not satisfy the determinedpassivity requirement, block S13 is implemented. Otherwise, if thenon-common-pole rational function satisfies the determined passivityrequirement, block S14 is directly implemented.

In block S13, the passivity enforcing module 314 enforces passivity ofthe non-common-pole rational function. In one embodiment, passivity ofthe non-common-pole rational function is enforced by converting thefunction (1) to the following function labeled (2):

$\begin{matrix}\begin{matrix}{\lbrack S\rbrack = \begin{bmatrix}{\sum\limits_{m = 1}^{M}\frac{r_{m}^{1,1}}{s + p_{m}^{1,1}}} & {\sum\limits_{m = 1}^{M}\frac{r_{m}^{1,2}}{s + p_{m}^{1,2}}} & \ldots & {\sum\limits_{m = 1}^{M}\frac{r_{m}^{1,N}}{s + p_{m}^{1,N}}} \\{\sum\limits_{m = 1}^{M}\frac{r_{m}^{2,1}}{s + p_{m}^{2,1}}} & {\sum\limits_{m = 1}^{M}\frac{r_{m}^{2,2}}{s + p_{m}^{2,2}}} & \ldots & {\sum\limits_{m = 1}^{M}\frac{r_{m}^{2,N}}{s + p_{m}^{2,N}}} \\\ldots & \ldots & \ldots & \ldots \\{\sum\limits_{m = 1}^{M}\frac{r_{m}^{1,N}}{s + p_{m}^{1,N}}} & {\sum\limits_{m = 1}^{M}\frac{r_{m}^{2,N}}{s + p_{m}^{2,N}}} & \ldots & {\overset{M}{\sum\limits_{m = 1}}\frac{r_{m}^{N,N}}{s + p_{m}^{N,N}}}\end{bmatrix}} \\{= {\sum\limits_{m = 1}^{M}{\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{i}{\frac{r_{m}^{i,j}}{s + p_{m}^{i,j}}{E_{i,j}.}}}}}}\end{matrix} & (2)\end{matrix}$In the function (2), E is a matrix may be shown using the followingfunction labeled (3):

$\begin{matrix}{{E_{u,v}( {i,j} )} = \{ {\begin{matrix}{1,} & {{i = ( {u\mspace{14mu}{or}\mspace{14mu} v} )},{j = {{( {u\mspace{14mu}{or}\mspace{14mu} v} )\mspace{14mu}{and}\mspace{14mu} i} \neq j}}} \\{0,} & {{o.w.},}\end{matrix}( {u \neq v} )} } & (3) \\{{E_{u,v}( {i,j} )} = \{ {\begin{matrix}{1,} & {i = {j = {u = v}}} \\{0,} & {{o.w.},}\end{matrix}{( {u = v} ).}} } & \;\end{matrix}$Using function (2), {tilde over (S)}, which represents modifiedS-parameters, may be found using the following function labeled (4):

$\begin{matrix}\begin{matrix}{\lfloor \overset{\sim}{S} \rfloor \cong {S + {\Delta\; S}}} \\{= {\sum\limits_{m = 1}^{M}{\sum\limits_{i = 1}^{N}\lbrack {\frac{E_{i,i}( {r_{m}^{i,i} + {\Delta\;\lambda_{m,i,i}}} )}{s + p_{m}^{i,i}} + {\sum\limits_{\underset{({j \neq i})}{j = 1}}^{i}{\frac{E_{i,j}( {r_{m}^{i,j} + {\Delta\;\lambda_{m,i,j}}} )}{s + p_{m}^{i,j}}.}}} }}}\end{matrix} & (4)\end{matrix}$In the function (4), Δλ is solved may be using the following functionslabeled (5a) and (5b) according to quadratic programming (QP):

$\begin{matrix}{{\min\limits_{\Delta\; x}{\frac{1}{2}( {\Delta\; x^{T\;}A_{sys}^{T}A_{sys}\Delta\; x} )}};} & ( {5a} ) \\{{B_{sys}\Delta\; x} < {c.}} & ( {5b} )\end{matrix}$Function (5a) is a least square equation. A_(sys) ^(T)A_(sys) infunction (5a) may be showed using the following function labeled (6):

$\begin{matrix}{{A_{sys}^{T}A_{sys}} = {\sum\limits_{k = 0}^{N_{s}}{{A_{base}( s_{k} )}^{T}{{A_{base}( s_{k} )}.}}}} & (6)\end{matrix}$In the function (6), s_(k) are different signal frequencies listed inthe S-parameter file 32.A_(base) (s_(k)) in the function (6) and Δx in the function (5a) may beshowed using the following functions labeled (7a) and (7b):

$\begin{matrix}{\mspace{20mu}{{A_{base}( s_{k} )} = \lbrack {{{\begin{matrix}{A_{base}^{1}( s_{k} )} & {A_{base}^{2}( s_{k} )} & \ldots & { {A_{base}^{M}( s_{k} )} \rbrack,}\end{matrix}\mspace{20mu}{A_{base}^{m}( s_{k} )}} = {A_{s}{A_{pole}^{m}( s_{k} )}}},\mspace{20mu}{{\Delta\; x} = \begin{bmatrix}{\Delta\; x_{1}} \\{\Delta\; x_{2}} \\\ldots \\{\Delta\; x_{M}}\end{bmatrix}},{{\Delta\; x_{m}} = {{\begin{bmatrix}{\Delta\;\lambda_{m,1,1}} \\{\Delta\;\lambda_{m,2,1}} \\{\Delta\;\lambda_{m,2,2}} \\\ldots \\{\Delta\;\lambda_{m,N,N}}\end{bmatrix}\mspace{20mu} A_{s}} = \begin{bmatrix}1 & 0 & 0 & \ldots & 0 \\0 & \sqrt{2} & 0 & \ldots & 0 \\0 & 0 & 1 & \ldots & 0 \\\ldots & \ldots & \ldots & \ldots & \ldots \\0 & 0 & 0 & \ldots & 1\end{bmatrix}}},;} }} & ( {7a} ) \\{{A_{pole}^{m}( s_{k} )} = {\begin{bmatrix}\frac{1}{s_{k} + p_{m}^{1,1}} & 0 & 0 & \ldots & 0 \\0 & \frac{1}{s_{k} + p_{m\;}^{2,1}} & 0 & \ldots & 0 \\0 & 0 & \frac{1}{s_{k} + p_{m}^{2,2}} & \ldots & 0 \\\ldots & \ldots & \ldots & \ldots & \ldots \\0 & 0 & 0 & \ldots & \frac{1}{s_{k} + p_{m}^{N,N}}\end{bmatrix}.}} & ( {7b} )\end{matrix}$Function (5b) is a constrain equation. B_(sys) and c in function (5b)may be showed using the following function labeled (8):

$\begin{matrix}{{B_{sys} = \begin{bmatrix}{\;{b_{k}( s_{1} )}} \\{b_{k}( s_{2} )} \\\ldots \\{b_{k}( s_{q} )}\end{bmatrix}},{c = {\begin{bmatrix}{\Delta\;\sigma_{k}^{s_{1}}} \\{\Delta\;\sigma_{k}^{s_{2}}} \\\ldots \\{\Delta\;\sigma_{k}^{s_{q}}}\end{bmatrix}.}}} & (8)\end{matrix}$In the function (8), Δσ_(k) ^(s) ^(q) is a modified value of anon-passivity system, k is the kth singular value of the S-parameters,s_(q) is a frequency of the local maximum singular value when k>1.The b_(k) (s) in function (8) may be showed using the following functionlabeled (9):b _(k)(s)=[u _(k) B _(base)(s)(I _(NexM){circle around (x)}ν_(k))]  (9).In the function (9), I_(N) _(c×M) is a (N_(c)×M)×(N_(c)×M) identitymatrix,

${N_{c} = {\frac{( {1 + N} )N}{2}M}},${circle around (x)} is Kronecker tensor product, u and v may be showedusing the following function labeled (10):

$\begin{matrix}{{U^{- 1} = \begin{bmatrix}u_{1} \\\ldots \\u_{N}\end{bmatrix}},{V = {\begin{bmatrix}v_{1} & \ldots & v_{N}\end{bmatrix}.}}} & (10)\end{matrix}$B_(base)(s) in function (9) may be showed using the following functionlabeled (11):B _(base)(s)=[B _(base) ¹(s)B _(base) ²(s) . . . B _(base) ^(M)(s)], B_(base) ^(m)(s)=B _(C) B _(pole) ^(m)(s)  (11).In the function (11), B_(c) and B_(pole) ^(m)(s) may be showed using thefollowing function labeled (12):

$\begin{matrix}{{B_{c} = \begin{bmatrix}E_{1,1} & E_{2,1} & E_{2,2} & \ldots & E_{N,N}\end{bmatrix}}{{B_{pole}^{m}(s)} = {\begin{bmatrix}\frac{I_{N}}{s + p_{m}^{1,1}} & 0 & 0 & \ldots & 0 \\0 & \frac{I_{N}}{s + p_{m}^{2,1}} & 0 & \ldots & 0 \\0 & 0 & \frac{I_{N}}{s + p_{m}^{2,2}} & \ldots & 0 \\0 & 0 & 0 & \ldots & \ldots \\0 & 0 & 0 & \ldots & \frac{I_{N}}{s + p_{m}^{N,N}}\end{bmatrix}.}}} & (12)\end{matrix}$In the function (12), I_(N) is a N×N identity matrix.

In block S14, the equivalent circuit generation module 315 generates anequivalent circuit model of the circuit 10 according to thenon-common-pole rational function adjusted in block S13 which satisfiesthe determined passivity requirement.

FIG. 4 is a flowchart of one embodiment detailing S12 in FIG. 3.Depending on the embodiment, additional blocks may be added, othersremoved, and the ordering of the blocks may be changed.

In block S120, the passivity analysis module 313 converts thenon-common-pole rational function to a state-space matrix. As mentionedabove, the non-common-pole rational function is the following function(1):

$\begin{matrix}{{{{S(s)} \approx {\hat{S}(s)}} = \begin{bmatrix}{{\hat{S}}_{11}(s)} & {{\hat{S}}_{12}(s)} & \ldots & {{\hat{S}}_{1N}(s)} \\{{\hat{S}}_{21}(s)} & {{\hat{S}}_{22}(s)} & \ldots & {{\hat{S}}_{2N}(s)} \\\ldots & \ldots & \ldots & \ldots \\{{\hat{S}}_{N\; 1}(s)} & {{\hat{S}}_{N\; 2}(s)} & \ldots & {{\hat{S}}_{NN}(s)}\end{bmatrix}},{wherein},{{{\hat{S}}_{ij}(s)} = {\sum\limits_{m = 1}^{M}{\frac{r_{m}^{i,j}}{s + p_{m}^{i,j}}.}}}} & (1)\end{matrix}$In one embodiment, the matrix conversion module 313 combines thefunction (1) and the following functions labeled (13a) and (13b) toobtain a combined matrix, then converts the combined matrix into thestate-space matrix using the following function labeled (14):

$\begin{matrix}{{\hat{S}{r_{p,q}(s)}} = {\sum\limits_{u = 1}^{U}\frac{r_{u}^{p,q}}{s + p_{u\;}^{p,q}}}} & ( {13a} )\end{matrix}$where p_(u) ^(p,q) and r_(u) ^(p,q) are real numbers.

$\begin{matrix}{{\hat{S}{c_{p,q}(s)}} = {{\sum\limits_{v = 1}^{V}\frac{{{Re}( r_{v}^{p,q} )} + {{{Im}( r_{v}^{p,q} )}j}}{s + {{Re}( p_{v\;}^{p,q} )} + {{{Im}( p_{v}^{p,q} )}j}}} + \frac{{{Re}( r_{v}^{p,q} )} - {{{Im}( r_{v}^{p,q} )}j}}{s + {{Re}( p_{v}^{p,q} )} - {{{Im}( p_{v}^{p,q} )}j}}}} & ( {13b} )\end{matrix}$where j=√{square root over (−1)}, U+2V=M, p_(u) ^(p,q)>0, and Re(p_(v)^(p,q))>0.jωX(jω)=AX(jω)+BU(jω)Y(jω)=CX(jω)+DU(jω),  (14).

In the function (14), A, B, C, and D may be showed using the followingfunction (15):

$\begin{matrix}{{A = \begin{bmatrix}A_{r} & 0 \\0 & A_{c}\end{bmatrix}},{B = \begin{bmatrix}B_{r} \\{B_{c}\;}\end{bmatrix}},{C = \begin{bmatrix}C_{r} & C_{c}\end{bmatrix}},{D = {\begin{bmatrix}d^{1,1} & d^{1,2} & \ldots & d^{1,N} \\d^{2,1} & d^{2,2} & \ldots & D^{2,N} \\\ldots & \ldots & \ldots & \ldots \\d^{N,1} & d^{N,1} & \ldots & d^{N,N}\end{bmatrix}.}}} & (15)\end{matrix}$

In the function (15), A_(r), B_(r) and C_(r) are state-space matrixes ofthe pole values Ŝr_(p,q)(s) in (13a), and may be showed using thefollowing function labeled (16):

$\begin{matrix}{{A_{r}( {i,j} )} = \{ \begin{matrix}{p_{u}^{p,q},} & {i = {{j\mspace{14mu}{and}\mspace{14mu} i} = {{( {U \cdot N} )( {p - 1} )} + {U( {q - 1} )} + u}}} \\0 & {o.w.}\end{matrix} } & (16) \\{{B_{r}( {i,j} )} = \{ \begin{matrix}{1,} & {i = {{{( {U \cdot N} )( {p - 1} )} + {U( {q - 1} )} + {u\mspace{14mu}{and}\mspace{14mu} j}} = q}} \\0 & {o.w.}\end{matrix} } & \; \\{{C_{r}( {i,j} )} = \{ \begin{matrix}{r_{u}^{p,q},} & {i = {{p\mspace{14mu}{and}\mspace{14mu} j} = {{( {U \cdot N} )( {p - 1} )} + {U( {q - 1} )} + u}}} \\0 & {{o.w}..}\end{matrix} } & \;\end{matrix}$

In the function (16), A_(r) is a (N·N·U)×(N·N·U) sparse matrix, B_(r) isa (N·U)×N sparse matrix, C_(r) is a N×(N·U) sparse matrix.

A_(c), B_(c), and C_(c) are state-space matrixes of the residue valuesŜc_(p,q)(s) in (13b), and may be showed using the following function(17):

$\begin{matrix}{{A_{c}( {i,j} )} = \{ \begin{matrix}{{{Re}( p_{v}^{p,q} )},} & {{i = {{j\mspace{14mu}{and}\mspace{14mu} i} = {\Psi( {p,q,v} )}}},{i = {{\Psi( {p,q,v} )} - 1}}} \\{{{Im}( p_{v}^{p,q} )},} & {{i = {{\Psi( {p,q,v} )} - 1}},{j = {\Psi( {p,q,v} )}}} \\{{- {{Im}( p_{v}^{p,q} )}},} & {{i = {\Psi( {p,q,v} )}},{j = {{\Psi( {p,q,v} )} - 1}}} \\{0,} & {o.w.}\end{matrix} } & (17) \\{\mspace{79mu}{{B_{c}( {i,j} )} = \{ \begin{matrix}{2,} & {i = {{{\Psi( {p,q,v} )} - {1\mspace{14mu}{and}\mspace{14mu} j}} = q}} \\0 & {o.w.}\end{matrix} }} & \; \\{\mspace{79mu}{{C_{c}( {i,j} )} = \{ \begin{matrix}{{{Re}( r_{v}^{p,q} )},} & {i = {{p\mspace{14mu}{and}\mspace{14mu} j} = {\Psi - 1}}} \\{{{Im}( r_{v}^{p,q} )},} & {i = {{p\mspace{14mu}{and}\mspace{14mu} j} = \Psi}} \\{0,} & {{o.w}..}\end{matrix} }} & \;\end{matrix}$

In the function (17), Ψ(p, q, v)=(2V·N)(p−1)+2V(q−1)+2v, A_(c) is asparse matrix, B_(r) is a (N·2V)×N sparse matrix, and C_(r) is aN×(N·2V) sparse matrix.

The matrix conversion module 313 combines the functions (15), (16), and(17) to obtain expressions of the coefficients A, B, C, and D in thestate-space matrix of the function (14).

In block S121, the passivity analysis module 313 substitutes thestate-space matrix of the function (14) into a Hamiltonian matrix, wherethe Hamiltonian matrix is the following function labeled (18):

$\begin{matrix}{H = \begin{bmatrix}{A - {{BR}^{- 1}D^{T}C}} & {{- {BR}^{- 1}}B^{T}} \\{C^{T}Q^{- 1}C} & {{- A^{T}} + {C^{T}{DR}^{- 1}B^{T}}}\end{bmatrix}} & (18)\end{matrix}$

where R=D^(T)D−I Q=DD^(T)−I, I is an identity matrix.

In block S309, the passivity analysis module 313 analyzes theeigenvalues of the Hamiltonian matrix for pure imaginaries, to determineif the non-common-pole rational function of the S-parameters satisfies adetermined passivity requirement. If the eigenvalues of the Hamiltonianmatrix have pure imaginaries, the passivity analysis module 313determines that the non-common-pole rational function of theS-parameters satisfies the determined passivity requirement. Otherwise,if the eigenvalues of the Hamiltonian matrix have no pure imaginaries,the passivity analysis module 313 determines that the non-common-polerational function of the S-parameters does not satisfy the determinedpassivity requirement.

Although certain inventive embodiments of the present disclosure havebeen specifically described, the present disclosure is not to beconstrued as being limited thereto. Various changes or modifications maybe made to the present disclosure without departing from the scope andspirit of the present disclosure.

What is claimed is:
 1. A computing device, comprising: a storage device;at least one processor; and a passivity enforcing unit comprising one ormore computerized codes, which are stored in the storage device andexecutable by the at least one processor, the one or more computerizedcodes comprising: a parameter reading module operable to read ascattering parameter (S-parameter) file from the storage device, whereinthe S-parameter file lists S-parameters measured at ports of a circuitat different signal frequencies; a vector fitting module operable tocreate a non-common-pole rational function of the S-parameters byapplying a vector fitting algorithm to the S-parameters; a passivityenforcing module operable to enforce passivity of the non-common-polerational function if the non-common-pole rational function does notsatisfy a determined passivity requirement; and an equivalent circuitgeneration module operable to generate an equivalent circuit model ofthe circuit according to the non-common-pole rational function whichsatisfies the determined passivity requirement by enforcing passivity.2. The computing device as claimed in claim 1, wherein the passivityenforcing module enforces passivity of the non-common-pole rationalfunction by the operation of: converting the non-common-pole rationalfunction which is showed using the function labeled (1) to a functionlabeled (2): $\begin{matrix}{{{{S(s)} \approx {\hat{S}(s)}} = \begin{bmatrix}{{\hat{S}}_{11}(s)} & {{\hat{S}}_{12}(s)} & \ldots & {{\hat{S}}_{1N}(s)} \\{{\hat{S}}_{21}(s)} & {{\hat{S}}_{22}(s)} & \ldots & {{\hat{S}}_{2N}(s)} \\\ldots & \ldots & \ldots & \ldots \\{{\hat{S}}_{N\; 1}(s)} & {{\hat{S}}_{N\; 2}(s)} & \ldots & {{\hat{S}}_{NN}(s)}\end{bmatrix}},{wherein},{{{\hat{S}}_{ij}(s)} = {\sum\limits_{m = 1}^{M}\frac{r_{m}^{i,j}}{s + p_{m}^{i,j}}}}} & (1) \\\begin{matrix}{\lbrack S\rbrack = \begin{bmatrix}{\sum\limits_{m = 1}^{M}\frac{r_{m}^{1,1}}{s + p_{m}^{1,1}}} & {\sum\limits_{m = 1}^{M}\frac{r_{m}^{1,2}}{s + p_{m}^{1,2}}} & \ldots & {\sum\limits_{m = 1}^{M}\frac{r_{m}^{1,N}}{s + p_{m}^{1,N}}} \\{\sum\limits_{m = 1}^{M}\frac{r_{m}^{2,1}}{s + p_{m}^{2,1}}} & {\sum\limits_{m = 1}^{M}\frac{r_{m}^{2,2}}{s + p_{m}^{2,2}}} & \ldots & {\sum\limits_{m = 1}^{M}\frac{r_{m}^{2,N}}{s + p_{m}^{2,N}}} \\\ldots & \ldots & \ldots & \ldots \\{\sum\limits_{m = 1}^{M}\frac{r_{m}^{1,N}}{s + p_{m}^{1,N}}} & {\sum\limits_{m = 1}^{M}\frac{r_{m}^{2,N}}{s + p_{m}^{2,N}}} & \ldots & {\sum\limits_{m = 1}^{M}\frac{r_{m}^{N,N}}{s + p_{m}^{N,N}}}\end{bmatrix}} \\{= {\sum\limits_{m = 1}^{M}{\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{i}{\frac{r_{m}^{i,j}}{s + p_{m}^{i,j}}{E_{i,j}.}}}}}}\end{matrix} & (2)\end{matrix}$ wherein M represents control precision of the function, Nrepresents the number of the ports of the circuit, r_(m) representsresidue values, p_(m) represents pole values, s =ω=2πƒ, ƒrepresents afrequency of a test signal, and E_(i,j) represents a matrix.
 3. Thecomputing device as claimed in claim 1, wherein the one or morecomputerized code further comprises: a passivity analysis moduleoperable to determine if the non-common-pole rational function satisfiesthe determined passivity requirement.
 4. The computing device as claimedin claim 3, wherein the passivity analysis module determines if thenon-common-pole rational function satisfies the determined passivityrequirement by the operations of: converting the non-common-polerational function to a state-space matrix; substituting the state-spacematrix into a Hamiltonian matrix, analyzing if eigenvalues of theHamiltonian matrix have pure imaginaries, and determining that thenon-common-pole rational function of the S-parameters satisfies adetermined passivity requirement if the eigenvalues of the Hamiltonianmatrix have pure imaginaries, or determining that the non-common-polerational function of the S-parameters does not satisfy the determinedpassivity requirement if the eigenvalues of the Hamiltonian matrix haveno pure imaginaries.
 5. A computer-based method for enforcing passivityof scattering parameter (S-parameter) equivalent circuit, the methodbeing performed by execution of computer readable program code by aprocessor of a computing device, and comprising: reading a S-parameterfile from a storage device, wherein the S-parameter file listsS-parameters measured at ports of a circuit at different signalfrequencies; creating a non-common-pole rational function of theS-parameters by applying a vector fitting algorithm to the S-parameters;enforcing passivity of the non-common-pole rational function if thenon-common-pole rational function does not satisfy a determinedpassivity requirement; and generating an equivalent circuit model of thecircuit according to the non-common-pole rational function whichsatisfies the determined passivity requirement by enforcing passivity.6. The method as claimed in claim 5, wherein passivity of thenon-common-pole rational function is enforced by: converting thenon-common-pole rational function which is showed using a functionlabeled (1) to a function labeled (2): $\begin{matrix}{{{{S(s)} \approx {\hat{S}(s)}} = \begin{bmatrix}{{\hat{S}}_{11}(s)} & {{\hat{S}}_{12}(s)} & \ldots & {{\hat{S}}_{1N}(s)} \\{{\hat{S}}_{21}(s)} & {{\hat{S}}_{22}(s)} & \ldots & {{\hat{S}}_{2N}(s)} \\\ldots & \ldots & \ldots & \ldots \\{{\hat{S}}_{N\; 1}(s)} & {{\hat{S}}_{N\; 2}(s)} & \ldots & {{\hat{S}}_{NN}(s)}\end{bmatrix}},{wherein},{{{\hat{S}}_{ij}(s)} = {\sum\limits_{m = 1}^{M}\frac{r_{m}^{i,j}}{s + p_{m}^{i,j}}}}} & (1) \\\begin{matrix}{\lbrack S\rbrack = \begin{bmatrix}{\sum\limits_{m = 1}^{M}\frac{r_{m}^{1,1}}{s + p_{m}^{1,1}}} & {\sum\limits_{m = 1}^{M}\frac{r_{m}^{1,2}}{{s + p_{m}^{1,2}}\;}} & \ldots & {\sum\limits_{m = 1}^{M}\frac{r_{m\;}^{1,N}}{s + p_{m}^{1,N}}} \\{\sum\limits_{m = 1}^{M}\frac{r_{m}^{2,1}}{s + p_{m}^{2,1}}} & {\sum\limits_{m = 1}^{M}\frac{r_{m}^{2,2}}{s + p_{m}^{2,2}}} & \ldots & {\sum\limits_{m = 1}^{M}\frac{r_{m}^{2,N}}{s + p_{m}^{2,N}}} \\\ldots & \ldots & \ldots & \ldots \\{\sum\limits_{m = 1}^{M}\frac{r_{m}^{1,N}}{s + p_{m}^{1,N}}} & {\sum\limits_{m = 1}^{M}\frac{r_{m}^{2,N}}{s + p_{m}^{2,N}}} & \ldots & {\sum\limits_{m = 1}^{M}\frac{r_{m}^{N,N}}{s + p_{m}^{N,N}}}\end{bmatrix}} \\{= {\overset{M}{\sum\limits_{m = 1}}{\sum\limits_{i = 1}^{N}{\sum\limits_{j = 1}^{i}{\frac{r_{m}^{i,j}}{s + p_{m}^{i,j}}{E_{i,j}.}}}}}}\end{matrix} & (2)\end{matrix}$ wherein M represents control precision of the function, Nrepresents the number of the ports of the circuit, r_(m) representsresidue values, p_(m) represents pole values, s =ω=2πƒ, ƒrepresents afrequency of a test signal, and E_(i,j) represents a matrix.
 7. Themethod as claimed in claim 5, before enforcing passivity of thenon-common-pole rational function further comprising: determining if thenon-common-pole rational function satisfies the determined passivityrequirement.
 8. The method as claimed in claim 7, wherein if thenon-common-pole rational function satisfies the determined passivityrequirement is determined by: converting the non-common-pole rationalfunction to a state-space matrix; substituting the state-space matrixinto a Hamiltonian matrix, analyzing if eigenvalues of the Hamiltonianmatrix have pure imaginaries, and determining that the non-common-polerational function of the S-parameters satisfies a determined passivityrequirement if the eigenvalues of the Hamiltonian matrix have pureimaginaries, or determining that the non-common-pole rational functionof the S-parameters does not satisfy the determined passivityrequirement if the eigenvalues of the Hamiltonian matrix have no pureimaginaries.
 9. A non-transitory computer readable medium storing a setof instructions, the set of instructions capable of being executed by aprocessor of a computing device to perform a method for enforcingpassivity of scattering parameter (S-parameter) equivalent circuit, themethod comprising: reading a S-parameter file from a storage device,wherein the S-parameter file lists S-parameters measured at ports of acircuit at different signal frequencies; creating a non-common-polerational function of the S-parameters by applying a vector fittingalgorithm to the S-parameters; enforcing passivity of thenon-common-pole rational function if the non-common-pole rationalfunction does not satisfy a determined passivity requirement; andgenerating an equivalent circuit model of the circuit according to thenon-common-pole rational function which satisfies the determinedpassivity requirement by enforcing passivity.
 10. The non-transitorycomputer readable medium as claimed in claim 9, wherein passivity of thenon-common-pole rational function is enforced by: converting thenon-common-pole rational function which is showed using a functionlabeled (1) to a function labeled (2): $\begin{matrix}{{{{S(s)} \approx {\hat{S}(s)}} = \lbrack \begin{matrix}{{\hat{S}}_{11}(s)} & {{\hat{S}}_{12}(s)} & \ldots & {{\hat{S}}_{1N}(s)} \\{{\hat{S}}_{21}(s)} & {{\hat{S}}_{22}(s)} & \ldots & {{\hat{S}}_{2N}(s)} \\\ldots & \ldots & \ldots & \ldots \\{{\hat{S}}_{N\; 1}(s)} & {{\hat{S}}_{N\; 2}(s)} & \ldots & {{\hat{S}}_{NN}(s)}\end{matrix} \rbrack},{wherein},{{{\hat{S}}_{ij}(s)} = {\sum\limits_{m = 1}^{M}\;\frac{r_{m}^{i,j}}{s + p_{m}^{i,j}}}}} & (1) \\{\lbrack S\rbrack = {\begin{bmatrix}{\sum\limits_{m = 1}^{M}\;\frac{r_{m}^{1,1}}{s + p_{m}^{1,1}}} & {\sum\limits_{m = 1}^{M}\;\frac{r_{m}^{1,2}}{s + p_{m}^{1,2}}} & \ldots & {\sum\limits_{m = 1}^{M}\;\frac{r_{m}^{1,N}}{s + p_{m}^{1,N}}} \\{\sum\limits_{m = 1}^{M}\;\frac{r_{m}^{2,1}}{s + p_{m}^{2,1}}} & {\sum\limits_{m = 1}^{M}\;\frac{r_{m}^{2,2}}{s + p_{m}^{2,2}}} & \ldots & {\sum\limits_{m = 1}^{M}\;\frac{r_{m}^{2,N}}{s + p_{m}^{2,N}}} \\\ldots & \ldots & \ldots & \ldots \\{\sum\limits_{m = 1}^{M}\;\frac{r_{m}^{1,N}}{s + p_{m}^{1,N}}} & {\sum\limits_{m = 1}^{M}\;\frac{r_{m}^{2,N}}{s + p_{m}^{2,N}}} & \ldots & {\sum\limits_{m = 1}^{M}\;\frac{r_{m}^{N,N}}{s + p_{m}^{N,N}}} \\\; & \; & \; & \;\end{bmatrix} = {\sum\limits_{m = 1}^{M}\;{\sum\limits_{i = 1}^{N}\;{\sum\limits_{j = 1}^{i}\;{\frac{r_{m}^{i,j}}{s + p_{m}^{i,j}}E_{i,j}}}}}}} & (2)\end{matrix}$ wherein M represents control precision of the function, Nrepresents the number of the ports of the circuit, r_(m) representsresidue values, p_(m) represents pole values, s =ω=2πƒ, ƒrepresentsfrequency of a test signal, and E_(i,j) represents a matrix.
 11. Thenon-transitory computer readable medium as claimed in claim 9, whereinthe method further comprises: determining if the non-common-polerational function satisfies the determined passivity requirement. 12.The non-transitory computer readable medium as claimed in claim 11,wherein if the non-common-pole rational function satisfies thedetermined passivity requirement is determined by: converting thenon-common-pole rational function to a state-space matrix; substitutingthe state-space matrix into a Hamiltonian matrix, analyzing ifeigenvalues of the Hamiltonian matrix have pure imaginaries, anddetermining that the non-common-pole rational function of theS-parameters satisfies a determined passivity requirement if theeigenvalues of the Hamiltonian matrix have pure imaginaries, ordetermining that the non-common-pole rational function of theS-parameters does not satisfy the determined passivity requirement ifthe eigenvalues of the Hamiltonian matrix have no pure imaginaries.